Burgers vector: Difference between revisions
en>Luckas-bot m r2.7.1) (Robot: Adding tr:Burgers vektörü |
just removed the whole problematic, possibly unoriginal paragraph; it doesn't actually seem that helpful (just seems to basically say very verbosely that if you enter units in meters you get meters back ...) |
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{| class="wikitable" align="right" style="margin-left:10px" width="250" | |||
!bgcolor=#e7dcc3 colspan=2|5-cubic honeycomb | |||
|- | |||
|bgcolor=#ffffff align=center colspan=2|(no image) | |||
|- | |||
|bgcolor=#e7dcc3|Type||[[List_of_regular_polytopes#Tessellations of Euclidean space|Regular 5-space honeycomb]] | |||
|- | |||
|bgcolor=#e7dcc3|Family||[[Hypercube honeycomb]] | |||
|- | |||
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| {4,3<sup>3</sup>,4}<BR>t<sub>0,5</sub>{4,3<sup>3</sup>,4}<BR>{4,3,3,3<sup>1,1</sup>}<BR>{4,3,4}x{∞}<BR>{4,3,4}x{4,4}<BR>{4,3,4}x{∞}<sup>2</sup><BR>{4,4}<sup>2</sup>x{∞}<BR>{∞}<sup>5</sup> | |||
|- | |||
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]s|| | |||
{{CDD|node_1|4|node|3|node|3|node|3|node|4|node}}<BR>{{CDD|node_1|4|node|3|node|3|node|3|node|4|node_1}}<BR>{{CDD|node_1|4|node|3|node|3|node|split1|nodes}}<BR>{{CDD|node_1|4|node|3|node|3|node|4|node|2|node_1|infin|node}}<BR>{{CDD|node_1|4|node|3|node|3|node|4|node_1|2|node_1|infin|node}} | |||
<BR>{{CDD|node_1|4|node|3|node|4|node|2|node_1|4|node|4|node}}<BR>{{CDD|node_1|4|node|3|node|4|node_1|2|node_1|4|node|4|node}} | |||
<BR>{{CDD|node_1|4|node|3|node|4|node|2|node_1|infin|node|2|node_1|infin|node}}<BR>{{CDD|node_1|4|node|3|node|4|node_1|2|node_1|infin|node|2|node_1|infin|node}} | |||
<BR>{{CDD|node_1|4|node|4|node|2|node_1|infin|node|2|node_1|infin|node|2|node_1|infin|node}}<BR>{{CDD|node_1|4|node|4|node_1|2|node_1|infin|node|2|node_1|infin|node|2|node_1|infin|node}} | |||
<BR>{{CDD|node_1|4|node|4|node|2|node_1|4|node|4|node|2|node_1|infin|node}} | |||
<BR>{{CDD|node_1|4|node|4|node_1|2|node_1|4|node|4|node|2|node_1|infin|node}} | |||
<BR>{{CDD|node_1|4|node|4|node_1|2|node_1|4|node|4|node_1|2|node_1|infin|node}} | |||
<BR>{{CDD|node_1|infin|node|2|node_1|infin|node|2|node_1|infin|node|2|node_1|infin|node|2|node_1|infin|node}} | |||
|- | |||
|bgcolor=#e7dcc3|5-face type||[[5-cube|{4,3<sup>3</sup>}]] | |||
|- | |||
|bgcolor=#e7dcc3|4-face type||[[tesseract|{4,3,3}]] | |||
|- | |||
|bgcolor=#e7dcc3|Cell type||[[cube|{4,3}]] | |||
|- | |||
|bgcolor=#e7dcc3|Face type||[[square (geometry)|{4}]] | |||
|- | |||
|bgcolor=#e7dcc3|Face figure||[[cube|{4,3}]]<BR>([[octahedron]]) | |||
|- | |||
|bgcolor=#e7dcc3|Edge figure||8 [[tesseract|{4,3,3}]]<BR>([[16-cell]]) | |||
|- | |||
|bgcolor=#e7dcc3|Vertex figure||32 [[5-cube|{4,3<sup>3</sup>}]]<BR>([[5-orthoplex]]) | |||
|- | |||
|bgcolor=#e7dcc3|[[Coxeter group]]||<math>{\tilde{C}}_5</math>, [4,3<sup>3</sup>,4] | |||
|- | |||
|bgcolor=#e7dcc3|Dual||[[Self-dual polytope|self-dual]] | |||
|- | |||
|bgcolor=#e7dcc3|Properties||[[vertex-transitive]], [[edge-transitive]], [[face-transitive]], [[cell-transitive]] | |||
|} | |||
The '''5-cubic honeycomb''' or '''penteractic honeycomb''' is the only regular space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]) in Euclidean 5-space. Four [[5-cube]]s meet at each cubic cell, and it is more explicitly called an ''order-4 penteractic honeycomb''. | |||
It is analogous to the [[square tiling]] of the plane and to the [[cubic honeycomb]] of 3-space, and the [[tesseractic honeycomb]] of 4-space. | |||
== Constructions== | |||
There are many different [[Wythoff construction]]s of this honeycomb. The most symmetric form is [[Regular polytope|regular]], with [[Schläfli symbol]] {4,3<sup>3</sup>,4}. Another form has two alternating [[5-cube]] facets (like a checkerboard) with Schläfli symbol {4,3,3,3<sup>1,1</sup>}. The lowest symmetry Wythoff construction has 32 types of facets around each vertex and a prismatic product Schläfli symbol {∞}<sup>5</sup>. | |||
== Related polytopes and honeycombs == | |||
The [4,3<sup>3</sup>,4], {{CDD|node|4|node|3|node|3|node|3|node|4|node}}, Coxeter group generates 63 permutations of uniform tessellations, 35 with unique symmetry and 34 with unique geometry. The [[Expansion (geometry)|expanded]] 5-cubic honeycomb is geometrically identical to the 5-cubic honeycomb. | |||
The ''5-cubic honeycomb'' can be [[Alternation (geometry)|alternated]] into the [[5-demicubic honeycomb]], replacing the 5-cubes with [[5-demicube]]s, and the alternated gaps are filled by [[5-orthoplex]] facets. | |||
It is also related to the regular [[6-cube]] which exists in 6-space with ''3'' ''5''-cubes on each cell. This could be considered as a tessellation on the [[N-sphere|5-sphere]], an ''order-3 penteractic honeycomb'', {4,3<sup>4</sup>}. | |||
=== Tritruncated 5-cubic honeycomb === | |||
A '''tritruncated 5-cubic honeycomb''', {{CDD|branch_11|3ab|nodes|4a4b|nodes}}, containins all [[bitruncated 5-orthoplex]] facets and is the [[Voronoi tessellation]] of the [[5-demicube honeycomb#D5 lattice|D<sub>5</sub><sup>*</sup> lattice]]. Facets can be identically colored from a doubled <math>{\tilde{C}}_5</math>×2, <nowiki>[[</nowiki>4,3<sup>3</sup>,4]] symmetry, alternately colored from <math>{\tilde{C}}_5</math>, [4,3<sup>3</sup>,4] symmetry, three colors from <math>{\tilde{B}}_5</math>, [4,3,3,3<sup>1,1</sup>] symmetry, and 4 colors from <math>{\tilde{D}}_5</math>, [3<sup>1,1</sup>,3,3<sup>1,1</sup>] symmetry. | |||
==See also== | |||
*[[List of regular polytopes]] | |||
Regular and uniform honeycombs in 5-space: | |||
*[[5-demicubic honeycomb]] | |||
*[[5-simplex honeycomb]] | |||
*[[Truncated 5-simplex honeycomb]] | |||
*[[Omnitruncated 5-simplex honeycomb]] | |||
== References == | |||
* [[Coxeter|Coxeter, H.S.M.]] ''[[Regular Polytopes (book)|Regular Polytopes]]'', (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 p. 296, Table II: Regular honeycombs | |||
* '''Kaleidoscopes: Selected Writings of H.S.M. Coxeter''', edited by F. Arthur Sherk, [[Peter McMullen]], Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html] | |||
** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', [Math. Zeit. 200 (1988) 3-45] | |||
{{Honeycombs}} | |||
[[Category:Honeycombs (geometry)]] | |||
[[Category:6-polytopes]] | |||
Revision as of 17:02, 14 September 2013
| 5-cubic honeycomb | |
|---|---|
| (no image) | |
| Type | Regular 5-space honeycomb |
| Family | Hypercube honeycomb |
| Schläfli symbol | {4,33,4} t0,5{4,33,4} {4,3,3,31,1} {4,3,4}x{∞} {4,3,4}x{4,4} {4,3,4}x{∞}2 {4,4}2x{∞} {∞}5 |
| Coxeter-Dynkin diagrams |
Template:CDD |
| 5-face type | {4,33} |
| 4-face type | {4,3,3} |
| Cell type | {4,3} |
| Face type | {4} |
| Face figure | {4,3} (octahedron) |
| Edge figure | 8 {4,3,3} (16-cell) |
| Vertex figure | 32 {4,33} (5-orthoplex) |
| Coxeter group | , [4,33,4] |
| Dual | self-dual |
| Properties | vertex-transitive, edge-transitive, face-transitive, cell-transitive |
The 5-cubic honeycomb or penteractic honeycomb is the only regular space-filling tessellation (or honeycomb) in Euclidean 5-space. Four 5-cubes meet at each cubic cell, and it is more explicitly called an order-4 penteractic honeycomb.
It is analogous to the square tiling of the plane and to the cubic honeycomb of 3-space, and the tesseractic honeycomb of 4-space.
Constructions
There are many different Wythoff constructions of this honeycomb. The most symmetric form is regular, with Schläfli symbol {4,33,4}. Another form has two alternating 5-cube facets (like a checkerboard) with Schläfli symbol {4,3,3,31,1}. The lowest symmetry Wythoff construction has 32 types of facets around each vertex and a prismatic product Schläfli symbol {∞}5.
Related polytopes and honeycombs
The [4,33,4], Template:CDD, Coxeter group generates 63 permutations of uniform tessellations, 35 with unique symmetry and 34 with unique geometry. The expanded 5-cubic honeycomb is geometrically identical to the 5-cubic honeycomb.
The 5-cubic honeycomb can be alternated into the 5-demicubic honeycomb, replacing the 5-cubes with 5-demicubes, and the alternated gaps are filled by 5-orthoplex facets.
It is also related to the regular 6-cube which exists in 6-space with 3 5-cubes on each cell. This could be considered as a tessellation on the 5-sphere, an order-3 penteractic honeycomb, {4,34}.
Tritruncated 5-cubic honeycomb
A tritruncated 5-cubic honeycomb, Template:CDD, containins all bitruncated 5-orthoplex facets and is the Voronoi tessellation of the D5* lattice. Facets can be identically colored from a doubled ×2, [[4,33,4]] symmetry, alternately colored from , [4,33,4] symmetry, three colors from , [4,3,3,31,1] symmetry, and 4 colors from , [31,1,3,31,1] symmetry.
See also
Regular and uniform honeycombs in 5-space:
- 5-demicubic honeycomb
- 5-simplex honeycomb
- Truncated 5-simplex honeycomb
- Omnitruncated 5-simplex honeycomb
References
- Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 p. 296, Table II: Regular honeycombs
- Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1]
- (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
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